Ab initio calculations of the mechanical and electronic properties of strained Si nanowires

This paper reports a systematic study of the mechanical and electronic properties of strained small diameter (0.7–2.6 nm) silicon nanowires (Si NWs) using ab initio density functional theory calculations. The values of Young’s modulus, Poisson ratio, band gap, effective mass, work function, and deformation potentials are calculated for $⟨110⟩$ and $⟨111⟩$ Si NWs. We find that quantum confinement in $⟨110⟩$ Si NWs splits conduction band valleys and decreases transport effective mass compared to the bulk case. Consequently, additional tensile strain should not lead to further significant electron mobility improvement. An interesting finding we report in this paper is that under compressive strain, there is a dramatic decrease in deformation potentials of $⟨110⟩$ Si NWs, which may result in a strong increase in electron mobilities, despite a concurrent increase in effective mass. We also observe a similar strain-induced counterplay of hole deformation potentials and effective masses for both $⟨110⟩$ and $⟨111⟩$ Si NWs. Finally, we do not see any significant effect of tensile or compressive strain on electron effective masses and deformation potentials in $⟨111⟩$ Si NWs. The sudden changes in effective mass and deformation potentials are concurrent with a change in the conduction and valence band edge states. In $⟨110⟩$ NWs, this change corresponds to a transition from direct-to-indirect band gap under strain.

Figure 1

(Color online) Cross section for (a) $⟨110⟩$ and (b) $⟨111⟩$ NWs. The labels of the NWs correspond to those in Table . The side view of the NWs about 2 nm in diameter are shown.

Figure 2

(Color online) Plot of (a) equilibrium axial lattice constant $L_0$ (% strain from bulk) and (b) Young’s modulus $Y$ (fraction of bulk Young’s modulus) versus diameter $d$ $\left(\text{Å}\right)$. The equilibrium axial lattice constant was fit by the curve $C_1/d+C_2/d^2$, and Young’s modulus was fit by $1-D_1/d-D_2/d^2$.

Figure 3

(Color online) Band structure for (a) $⟨110⟩$ and (b) $⟨111⟩$ NWs about 2 nm in size (wires 3 and 8) under different amounts of strain.

Figure 4

(Color online) Electronic charge density of the conduction and valence band edge states for (a) the $⟨110⟩$ NW and (b) the $⟨111⟩$ NW. The states are labeled as in Fig. 3. The $XY$ plane (middle), $YZ$ plane (left), and $XZ$ plane (top) are shown for each state where the charge is averaged along the out of plane axis. High density is indicated as red and low density is indicated as blue. The H atoms are also shown in the plots to give an idea of where the electronic charge is present spatially in the NW.

Figure 5

(Color online) Band gap versus strain for $⟨110⟩$ and $⟨111⟩$ Si NWs of different sizes.

Figure 6

(Color online) Electron and hole effective masses versus strain of Si NWs. (a) shows the electron effective mass versus strain of $⟨110⟩$ Si NWs of different sizes. The $⟨110⟩$ NWs undergo a direct-to-indirect band gap transition under compressive strain along with an associated jump in the electron effective mass. This happens at $-7\%$, $-4\%$, $-3\%$, and $-3\%$ for wires 1, 2, 3, and 4 respectively. (b) shows the hole effective mass versus strain of $⟨110⟩$ Si NWs of different sizes. The valence bands also undergo a $\Gamma$ to non-$\Gamma$ transition under tensile strain at 10%, 4%, 3%, and 2% for wires 1, 2, 3, and 4 respectively. (c) shows the hole effective mass versus strain of $⟨111⟩$ Si NWs of different sizes. The $⟨111⟩$ Si NWs valence edge state also transitions away from $\Gamma$ at 9% and 6% tensile strain for NWs 7 and 8, respectively.

Figure 7

(Color online) WF versus strain data and analysis for NWs. (a) displays the WF versus strain for $⟨110⟩$ and $⟨111⟩$ Si NWs of different sizes. The bulk Si WF of 4.2 eV is also plotted for comparison. (b) and (c) give a quantitative analysis of the surface dipole contribution to WF change in Si NWs. (b) displays the change in average Si atom potential with respect to vacuum, $V_{\text{ref}}-\varphi$, plotted as a function of strain. (c) shows the change in charge density of NW 3 after 1% tensile strain. Red, orange, and yellow indicate positive charge transfer, while green, blue, indigo, and purple indicate negative charge transfer. The figure shows that dipole is reduced with increasing tension, which signifies the increase in $V_{\text{ref}}$ in (b).